This article is a project that takes students through the process of forming a mathematical model of bicycle dynamics. Beginning with basic ideas from Newtonian mechanics (forces and torques), students use techniques from calculus and differential equations to develop the equations of rotational motion for a bicycle-rider system as it tips from side to side. Then students are led through a self-discovery process which uses a computer model to test various attempts to "ride" the bicycle so as to keep it upright and traveling along a straight path. (Contains 9 figures.)